Theorem imass1 5948

Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
imass1	⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Proof of Theorem imass1

Step	Hyp	Ref	Expression
1		ssres 5862	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))
2		rnss 5792	. . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
3	1, 2	syl 17	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
4		df-ima 5548	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
5		df-ima 5548	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
6	3, 4, 5	3sstr4g 3932	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3853  ran crn 5536   ↾ cres 5537   “ cima 5538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548

This theorem is referenced by:  vdwnnlem1  16443  dprdres  19281  imasnopn  22453  imasncld  22454  imasncls  22455  utoptop  22998  restutop  23001  ustuqtop3  23007  utopreg  23016  metustbl  23331  imadifxp  30526  esum2d  31643  eulerpartlemmf  31924  bj-imdirco  35014  brtrclfv2  40921  frege97d  40946  frege109d  40951  frege131d  40958  hess  40974  resimass  42361  setrecsss  45906